Imaginary. Axis. Real. Axis


 Philomena Snow
 7 months ago
 Views:
Transcription
1 Name ME6 Final. I certify that I upheld the Stanford Honor code during this exam Monday December 2, 25 3:36:3 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes, homework, books, etc. ffl Write your answers on this handout ffl Where space is provided, show your work to get credit ffl If necessary, attach extra pages for scratch work ffl Best wishes for a fun vacation. Merry Christmas and Happy New Year! Page Value Score Total F. (2 pts.) Miscellaneous (a) ( pt.) What is your MeyerBriggs (human metrics) personality type? (b) ( pt.) Suppose you plan on teaching or TAing Dynamic Systems. Provide suggestions for improving the course. Specifically, what you would add, remove, or change in the course (e.g., content, homework, book, lectures, etc.)?
2 F.2 (3 pts.) Coupled motions of Wilburforce pendulum N N o Shown to the right is a rigid body B that is attached to a spring at B c, the mass center of B. The other end of the spring is attached to point N o which is fixed in a Newtonian reference frame N. The equations governing this model of the system are B m ẍ + k x x + k c = I + k c x + k = B c Quantity Symbol Type Mass of B m constant Central moment of inertia of B about vertical I constant Linear spring constant modeling extensional flexibility k x constant Linear spring constant modeling torsional flexibility k constant Linear spring constant modeling coupled flexibility k c constant Translational stretch of spring from equilibrium x dependent variable Rotational stretch of spring from equilibrium dependent variable (a) (2 pts.) Write the ODEs in the matrix form M Ẍ + B _X + KX =, where X =» x» m I» ẍ +»» _x _ + " k x k c k c k #» x (b) (9 pts.) The solution to the previous set of ODEs has the form X(t)=U Λe pt where p is a constant (tobedetermined) and U is a nonzero 2 matrix of constants (tobedetermined). Fill in the two blanks in the following polynomial equation that governs the values of =  p 2. Note: the blanks only involve m, I, k x, k, k c. =» kx m + k I Λ + k x k k 2 c mi = 2
3 (c) (6 pts.) For certain values of m, I, k x, k, and k c, the matrix A = M  K = Find the eigenvalues and corresponding eigenvectors of A.». = 9 U =»  2 = U 2 =» (d) (2 pts.) Calculate the values of p, p 2, p 3, p 4 and draw their locations in the complex plane. p ;2 = ± p  = ± p 9 = ± 3 i p 3;4 = ± p  2 = ± p  = ± 3:32 i Imaginary Axis x x Real Axis (e) ( pt.) The solution is stable/ neutrally stable /unstable. (f) (4 pts.) Write the solution for X(t) in terms of the yettobedetermined constants c, c 2, c 3, c 4, the sine and cosine functions, and t.» x(t) (t) =»  n c sin( 3 t) + c 2 cos( 3 t)o 3 +» n c 3 sin( 3:32 t) + c 4 cos( 3:32 t)o
4 (g) (2 pts.) Determine c, c 2, c 3, c 4 when x() = :2, ()=, _x() =, and _ () =. c = c 2 = : c 3 = c 4 = : (h) (2 pts.) Using the aforementioned initial values, write explicit solutions for x(t) and (t). x(t) = : cos(3 t) + : cos(3:32 t) (t) = : cos(3 t) + : cos(3:32 t) (i) ( pt.) Using trigonometric identities, equation (2.9), and cos(a) =cos(a+ß), show that your previous solution for x(t) and (t) is x(t) = :2 sin(:6 t + ß 2 ) sin(3:6 t + ß 2 ) = :2 cos(:6 t) cos(3:6 t).2.2 (t) = :2 sin(:6 t) sin(3:6 t) = :2 sin(:6 t + ß) sin(3:6 t).5.5 Displacement Rotation angle (radians) (j) (2 pts.) Give an interpretation of the timebehavior of x(t) and (t). Both x(t)and (t)exhibit the beat phenomena with a highfrequency of 3:6 rad sec rad and a lowfrequency of :6 sec. Since x(t)and (t)are coupled, x's maximum amplitude coincides with 's minimum amplitude (and viceversa). 4
5 F.3 (38 pts.) Linearization of equations of motion for a swinging spring A straight, massless, linear spring connects a particle Q to a point N o which is fixed in a Newtonian reference frame N. The system identifiers and governing equations are shown below. Quantity Identifier Type Local gravitational constant g constant Mass of Q m constant Natural spring length L n constant Linear spring constant k constant Spring stretch y variable Pendulum angle" variable m (L n + y) + 2 m _ _y + mg sin( ) = m ÿ m (L n +y) _ 2 + ky mg cos( ) = N N o b y k θ L n n z n y n x y b x (a) ( pt.) Classify the previous equations by picking the relevant qualifiers from the following list. Uncoupled Linear Homogeneous Constantcoefficient storder Algebraic Coupled Nonlinear Inhomogeneous Variablecoefficient 2ndorder Differential (b) (3 pts.) Find the condition on y nom such that y = y nom (a constant) and = nom = satisfy the nonlinear ODEs. y nom = mg k (c) ( pts.) Using a Taylor series, linearize the differential equations in perturbations about this nominal solution. In other words, define perturbations of y and as ey = y y nom (y nom is constant) e = nom ( nom =) and form a set of differential equations which arelineariney, _ ey, ëy, e, _ e, ë. m (L n + y nom ) ë + mg e = 5 m ëy + k ey =
6 (d) (7 pts.) Replace y and in the original nonlinear ODEs with y = ey + y nom (y nom is the constant determined earlier) = e + nom ( nom =) Assuming ey, _ ey, ëy and e, _ e, and ë are small and using the smallapproximations technique of Chapter 2, form a set of ODEs that are linear in ey, _ ey, ëy, e, _ e, ë. m (L n + y nom ) ë + mg e = m ëy + k ey = (e) ( pt.) Classify your equations in part (3d) by picking the relevant qualifiers from the following list. Uncoupled Linear Homogeneous Constantcoefficient storder Algebraic Coupled Nonlinear Inhomogeneous Variablecoefficient 2ndorder Differential (f) ( pt.) The linearized ODEs resulting from the Taylor series technique are identical to those obtained from the smallapproximations technique. True /False. (g) (5 pts.) Find analytical solutions for ey(t) and (t) e when g =9:8 m=sec 2, m = kg, L n =:5 m, k = n=m, ey() = : m, e () = ffi,and _ ey() = _ e () = ey(t) = e (t) = q k : cos( t) m = : cos( t) q q ffi g cos( L n+y t) nom = ffi cos( gk t) = L n k+mg ffi cos(4:5 t) 6
7 (h) (4 pts.) Make a rough sketch of your linearized solutions for y(t)and (t)for» t» 6 sec, showing the relevant characteristics, e.g., amplitude, frequency, growth, decay, etc. y(t) ß ey(t)+y nom = : cos( t)+:98 (t) ß e (t)+ nom = ffi cos(4:5 t).2 3 Spring stretch (meters).5..5 Pendulum angle (degrees) (i) (6 pts.) The computer solution for y(t) and (t) to the original nonlinear ODEs for» t» 6 sec are plotted below. Comment on the similarities and differences between the plots you produced with the ones below. Give a reason for the similarities and differences..2 3 Spring stretch (meters) Similarities: Frequency, maximum amplitude Pendulum angle (degrees) Similarities: Frequency Differences: Beat phenomenon, varying amplitude Differences: Amplitude, beat phenomenon Reason for differences: Coupled and nonlinear Reason for differences: Coupled and nonlinear 7
8 F.4 (29 pts.) Dynamic response for an airconditioner on a building An air conditioner is bolted to the roof of a one story building. The air conditioner's motor is unbalanced and its eccentricity is modeled as a particle of mass m attached to the distal end of a rigid rod of length r. When the motor spins with angular speed Ω, it causes the building's roof of mass M to vibrate. The stiffness and material damping in each column that supports the roof is modeled as a linear horizontal spring (k) and linear horizontal damper (b). Homework 4.5 showed that the ODE governing the horizontal displacement x of the building's roof is (M +m)ẍ + 2 b _x + 2 kx = mrω 2 sin(ωt) (a) (3 pts.) Express! n,, A in terms of M, m, b, k, r so the ODE is x k,b M Ωt r m k,b ẍ + 2! n _x +! 2 n x = AΩ2 sin(ωt)! n = q 2 k b = p M + m (M+m) k A = mr M + m (b) (2 pts.) For certain values of M, m, b, k, andr, this ODE simplifies to Calculate numerical values for! n and. ẍ + 3 _x + 9 x = x 4 Ω 2 sin(ωt)! n = 3 rad sec = :5 (c) (7 pts.) Fillinnumerical valuesinthefollowing expressions for x ss (t), the steadystate part of x(t). Ω ( rad sec ) x ss(t) 2 7:94 x 5 Λ sin( 2 t + 6:843 ffi ) 3 : Λ sin( 3 t + 9 ffi ) 4 2:25 x 4 Λ sin( 4 t + 7:3 ffi ) 8
9 (d) (4 pts.) The building's occupants complain that the roof shakes too much. Comment on Ω the effect small variations of M, m, b, k, and r have on:,! n, and the magnitude of x ss (t). Fill in each element in the table by writing (decreases), (no effect), + (increases), or? (if it may decrease or increase). For the 2 nd tolast column, assume the air conditioner's normal operating speed is Ω = 28 rad rad, and for last column, assume Ω = 32. sec sec Ω!n Ω ß 28 rad sec jx ss (t)j Ω ß 32 rad sec jx ss (t)j Balancing the motor (r! ) Increasing the motor speed Ω + + Decreasing the motor speed Ω + Adding mass to the roof (increasing M) + +?? Removing mass from the roof +?? Stiffening the support columns (increasing k) + Adding damping to the columns (increasing b) + (e) (3 pts.) List three ways to change the motor and minimize the roof shaking. Balance the motor, damp the motor, or operate Ω fl! n or Ω fi! n. 9
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1  TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1  TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationModeling and Experimentation: MassSpringDamper System Dynamics
Modeling and Experimentation: MassSpringDamper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to
More informationEquations. A body executing simple harmonic motion has maximum acceleration ) At the mean positions ) At the two extreme position 3) At any position 4) he question is irrelevant. A particle moves on the
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves JanFeb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office
More informationAssignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class
Assignments VIII and IX, PHYS 301 (Classical Mechanics) Spring 2014 Due 3/21/14 at start of class Homeworks VIII and IX both center on Lagrangian mechanics and involve many of the same skills. Therefore,
More informationAP Physics. Harmonic Motion. Multiple Choice. Test E
AP Physics Harmonic Motion Multiple Choice Test E A 0.10Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 18. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More information5.6 Unforced Mechanical Vibrations
5.6 Unforced Mechanical Vibrations 215 5.6 Unforced Mechanical Vibrations The study of vibrating mechanical systems begins here with examples for unforced systems with one degree of freedom. The main example
More informationInvestigating Springs (Simple Harmonic Motion)
Investigating Springs (Simple Harmonic Motion) INTRODUCTION The purpose of this lab is to study the wellknown force exerted by a spring The force, as given by Hooke s Law, is a function of the amount
More informationPHY217: Vibrations and Waves
Assessed Problem set 1 Issued: 5 November 01 PHY17: Vibrations and Waves Deadline for submission: 5 pm Thursday 15th November, to the V&W pigeon hole in the Physics reception on the 1st floor of the GO
More informationPLANAR KINETICS OF A RIGID BODY: WORK AND ENERGY Today s Objectives: Students will be able to: 1. Define the various ways a force and couple do work.
PLANAR KINETICS OF A RIGID BODY: WORK AND ENERGY Today s Objectives: Students will be able to: 1. Define the various ways a force and couple do work. InClass Activities: 2. Apply the principle of work
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = kx When the mass is released, the spring will pull
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A 4.8kg block attached to a spring executes simple harmonic motion on a frictionless
More informationMath Assignment 5
Math 2280  Assignment 5 Dylan Zwick Fall 2013 Section 3.41, 5, 18, 21 Section 3.51, 11, 23, 28, 35, 47, 56 Section 3.61, 2, 9, 17, 24 1 Section 3.4  Mechanical Vibrations 3.4.1  Determine the period
More informationHonors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using tbar method. A. Sine and Cosecant. y = sinθ y y y y 0   80   30 0 0 300 5 35 5 35 60 50 0
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More information4.5. Applications of Trigonometry to Waves. Introduction. Prerequisites. Learning Outcomes
Applications of Trigonometry to Waves 4.5 Introduction Waves and vibrations occur in many contexts. The water waves on the sea and the vibrations of a stringed musical instrument are just two everyday
More informationFALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003
FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003 NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 14 pages. Make sure none are missing 2. There is
More informationEECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators.
Name: SID: EECS C28/ ME C34 Final Wed. Dec. 5, 2 8 am Closed book. Two pages of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 2 2 6 3 4 4 5 6 6 7 8 2 Total
More informationThe Phasor Analysis Method For Harmonically Forced Linear Systems
The Phasor Analysis Method For Harmonically Forced Linear Systems Daniel S. Stutts, Ph.D. April 4, 1999 Revised: 1015010, 91011 1 Introduction One of the most common tasks in vibration analysis is
More informationHealy/DiMurro. Vibrations 2016
Name Vibrations 2016 Healy/DiMurro 1. In the diagram below, an ideal pendulum released from point A swings freely through point B. 4. As the pendulum swings freely from A to B as shown in the diagram to
More informationChapter 7 Hooke s Force law and Simple Harmonic Oscillations
Chapter 7 Hooke s Force law and Simple Harmonic Oscillations Hooke s Law An empirically derived relationship that approximately works for many materials over a limited range. Exactly true for a massless,
More information4. Sinusoidal solutions
16 4. Sinusoidal solutions Many things in nature are periodic, even sinusoidal. We will begin by reviewing terms surrounding periodic functions. If an LTI system is fed a periodic input signal, we have
More informationOscillations Simple Harmonic Motion
Oscillations Simple Harmonic Motion Lana Sheridan De Anza College Dec 1, 2017 Overview oscillations simple harmonic motion (SHM) spring systems energy in SHM pendula damped oscillations Oscillations and
More informationExamination paper for TMA4195 Mathematical Modeling
Department of Mathematical Sciences Examination paper for TMA4195 Mathematical Modeling Academic contact during examination: Elena Celledoni Phone: 48238584, 73593541 Examination date: 11th of December
More information18.12 FORCEDDAMPED VIBRATIONS
8. ORCEDDAMPED VIBRATIONS Vibrations A mass m is attached to a helical spring and is suspended from a fixed support as before. Damping is also provided in the system ith a dashpot (ig. 8.). Before the
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More informationRobotics. Dynamics. Marc Toussaint U Stuttgart
Robotics Dynamics 1D point mass, damping & oscillation, PID, dynamics of mechanical systems, EulerLagrange equation, NewtonEuler recursion, general robot dynamics, joint space control, reference trajectory
More informationContents. Dynamics and control of mechanical systems. Focus on
Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies
More informationLab 11  Free, Damped, and Forced Oscillations
Lab 11 Free, Damped, and Forced Oscillations L111 Name Date Partners Lab 11  Free, Damped, and Forced Oscillations OBJECTIVES To understand the free oscillations of a mass and spring. To understand how
More informationUndamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator)
Section 3. 7 MassSpring Systems (no damping) Key Terms/ Ideas: Hooke s Law of Springs Undamped Free Vibrations (Simple Harmonic Motion; SHM also called Simple Harmonic Oscillator) Amplitude Natural Frequency
More informationGood Vibes: Introduction to Oscillations
Good Vibes: Introduction to Oscillations Description: Several conceptual and qualitative questions related to main characteristics of simple harmonic motion: amplitude, displacement, period, frequency,
More informationFriction. Modeling, Identification, & Analysis
Friction Modeling, Identification, & Analysis Objectives Understand the friction phenomenon as it relates to motion systems. Develop a controloriented model with appropriate simplifying assumptions for
More informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationExam 3 Results !"#$%&%'()*+(,./0% 123+#435%%6789:% Approximate Grade Cutoffs Ø A Ø B Ø C Ø D Ø 0 24 F
Exam 3 Results Approximate Grade Cutos Ø 751 A Ø 55 74 B Ø 35 54 C Ø 5 34 D Ø 4 F '$!" '#!" '!!" &!" %!" $!" #!"!"!"#$%&%'()*+(,./% 13+#435%%6789:%!()" )('!" '!(')" ')(#!" #!(#)" #)(*!" *!(*)" *)($!"
More informationPHYSICS 1 Simple Harmonic Motion
Advanced Placement PHYSICS Simple Harmonic Motion Presenter 0405 Simple Harmonic Motion What I Absolutely Have to Know to Survive the AP* Exam Whenever the acceleration of an object is proportional to
More informationFigure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody
Lecture 27. THE COMPOUND PENDULUM Figure 5.16 Compound pendulum: (a) At rest in equilibrium, (b) General position with coordinate θ, Freebody diagram The term compound is used to distinguish the present
More informationEXAMPLE 2: CLASSICAL MECHANICS: Worked examples. b) Position and velocity as integrals. Michaelmas Term Lectures Prof M.
CLASSICAL MECHANICS: Worked examples Michaelmas Term 2006 4 Lectures Prof M. Brouard EXAMPLE 2: b) Position and velocity as integrals Calculate the position of a particle given its time dependent acceleration:
More informationMath 312 Lecture Notes Linear Twodimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Twodimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More information2.003 Engineering Dynamics Problem Set 10 with answer to the concept questions
.003 Engineering Dynamics Problem Set 10 with answer to the concept questions Problem 1 Figure 1. Cart with a slender rod A slender rod of length l (m) and mass m (0.5kg)is attached by a frictionless pivot
More informationExam 3 Practice Solutions
Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at
More informationLab 14  Simple Harmonic Motion and Oscillations on an Incline
Lab 14  Simple Harmonic Motion and Oscillations on an Incline Name I. Introduction/Theory Partner s Name The purpose of this lab is to measure the period of oscillation of a spring and mass system on
More informationkg C 10 C = J J = J kg C 20 C = J J = J J
Seat: PHYS 1500 (Spring 2007) Exam #3, V1 Name: 5 pts 1. A pendulum is made with a length of string of negligible mass with a 0.25 kg mass at the end. A 2nd pendulum is identical except the mass is 0.50
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More information7. FORCE ANALYSIS. Fundamentals F C
ME 352 ORE NLYSIS 7. ORE NLYSIS his chapter discusses some of the methodologies used to perform force analysis on mechanisms. he chapter begins with a review of some fundamentals of force analysis using
More informationAPPLICATIONS OF SECONDORDER DIFFERENTIAL EQUATIONS
APPLICATIONS OF SECONDORDER DIFFERENTIAL EQUATIONS Secondorder linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration
More informationPhysics 351, Spring 2015, Homework #5. Due at start of class, Friday, February 20, 2015 Course info is at positron.hep.upenn.
Physics 351, Spring 2015, Homework #5. Due at start of class, Friday, February 20, 2015 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page
More informationChapter 1. Harmonic Oscillator. 1.1 Energy Analysis
Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the massspring system. A mass is attached to one end of a spring. The other end of the spring is attached to
More informationFinal Exam Solution Dynamics :45 12:15. Problem 1 Bateau
Final Exam Solution Dynamics 2 191157140 31012013 8:45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat
More informationa x b x b y b z Homework 3. Chapter 4. Vector bases and rotation matrices
Homework 3. Chapter 4. Vector bases and rotation matrices 3.1 Dynamics project rotation matrices Update your typed problem description and identifiers as necessary. Include schematics showing the bodies,
More informationSolution Derivations for Capa #12
Solution Derivations for Capa #12 1) A hoop of radius 0.200 m and mass 0.460 kg, is suspended by a point on it s perimeter as shown in the figure. If the hoop is allowed to oscillate side to side as a
More informationChapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion
Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular
More informationLAST TIME: Simple Pendulum:
LAST TIME: Simple Pendulum: The displacement from equilibrium, x is the arclength s = L. s / L x / L Accelerating & Restoring Force in the tangential direction, taking cw as positive initial displacement
More informationPHYS 1303 Final Exam Example Questions
PHYS 1303 Final Exam Example Questions (In summer 2014 we have not covered questions 3035,40,41) 1.Which quantity can be converted from the English system to the metric system by the conversion factor
More informationPHYSICS 218 FINAL EXAM Friday, December 11, 2009
PHYSICS 218 FINAL EXAM Friday, December 11, 2009 NAME: SECTION: 525 526 527 528 Note: 525 Recitation Wed 9:1010:00 526 Recitation Wed 11:3012:20 527 Recitation Wed 1:502:40 528 Recitation Mon 11:3012:20
More informationBrenda Rubenstein (Physics PhD)
PHYSICIST PROFILE Brenda Rubenstein (Physics PhD) Postdoctoral Researcher Lawrence Livermore Nat l Lab Livermore, CA In college, Brenda looked for a career path that would allow her to make a positive
More informationG r a d e 1 1 P r e  C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam
G r a d e 1 1 P r e  C a l c u l u s M a t h e m a t i c s ( 3 0 S ) Final Practice Exam G r a d e 1 1 P r e  C a l c u l u s M a t h e m a t i c s Final Practice Exam Name: Student Number: For Marker
More informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 56: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
More informationIntroduction to Vibration. Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high
More information5.1: Graphing Sine and Cosine Functions
5.1: Graphing Sine and Cosine Functions Complete the table below ( we used increments of for the values of ) 4 0 sin 4 2 3 4 5 4 3 7 2 4 2 cos 1. Using the table, sketch the graph of y sin for 0 2 2. What
More informationIntroduction to Mechanical Vibration
2103433 Introduction to Mechanical Vibration Nopdanai Ajavakom (NAV) 1 Course Topics Introduction to Vibration What is vibration? Basic concepts of vibration Modeling Linearization SingleDegreeofFreedom
More informationPhysics 231 Lecture 18
Physics 31 ecture 18 τ = Fd;d is the lever arm Main points of today s lecture: Energy Pendulum T = π g ( ) θ = θmax cos πft + ϑ0 Damped Oscillations x x equibrium = Ae bt/(m) cos(ω damped t) ω damped =
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationPhysics 202 Homework 1
Physics 202 Homework Apr 3, 203. A person who weighs 670 newtons steps onto a spring scale in the bathroom, (a) 85 kn/m (b) 290 newtons and the spring compresses by 0.79 cm. (a) What is the spring constant?
More information100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =
1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot
More informationGraded and supplementary homework, Math 2584, Section 4, Fall 2017
Graded and supplementary homework, Math 2584, Section 4, Fall 2017 (AB 1) (a) Is y = cos(2x) a solution to the differential equation d2 y + 4y = 0? dx2 (b) Is y = e 2x a solution to the differential equation
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. BernoulliEuler Beams.
More informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums MassSpring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationPhysics 106 Group Problems Summer 2015 Oscillations and Waves
Physics 106 Group Problems Summer 2015 Oscillations and Waves Name: 1. (5 points) The tension in a string with a linear mass density of 0.0010 kg/m is 0.40 N. What is the frequency of a sinusoidal wave
More informationMATH2351 Introduction to Ordinary Differential Equations, Fall Hints to Week 07 Worksheet: Mechanical Vibrations
MATH351 Introduction to Ordinary Differential Equations, Fall 111 Hints to Week 7 Worksheet: Mechanical Vibrations 1. (Demonstration) ( 3.8, page 3, Q. 5) A mass weighing lb streches a spring by 6 in.
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module  2 Simpul Rotors Lecture  2 Jeffcott Rotor Model In the
More informationTrigonometry Exam 2 Review: Chapters 4, 5, 6
Trig Exam Review F07 O Brien Trigonometry Exam Review: Chapters,, 0% of the questions on Exam will come from Chapters through. The other 70 7% of the exam will come from Chapters through. There may be
More informationWAVES & SIMPLE HARMONIC MOTION
PROJECT WAVES & SIMPLE HARMONIC MOTION EVERY WAVE, REGARDLESS OF HOW HIGH AND FORCEFUL IT CRESTS, MUST EVENTUALLY COLLAPSE WITHIN ITSELF.  STEFAN ZWEIG What s a Wave? A wave is a wiggle in time and space
More informationChapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.
Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 101 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 101 Angular Position, Velocity,
More informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More informationPES Physics 1 Practice Questions Exam 2. Name: Score: /...
Practice Questions Exam /page PES 0 003  Physics Practice Questions Exam Name: Score: /... Instructions Time allowed for this is exam is hour 5 minutes... multiple choice (... points)... written problems
More informationInstructor: Biswas/Ihas/Whiting PHYSICS DEPARTMENT PHY 2053 Exam 3, 120 minutes December 12, 2009
77777 77777 Instructor: Biswas/Ihas/Whiting PHYSICS DEPARTMENT PHY 2053 Exam 3, 120 minutes December 12, 2009 Name (print, last first): Signature: On my honor, I have neither given nor received unauthorized
More informationC. points X and Y only. D. points O, X and Y only. (Total 1 mark)
Grade 11 Physics  Homework 16  Answers on a separate sheet of paper, please 1. A cart, connected to two identical springs, is oscillating with simple harmonic motion between two points X and Y that
More informationVibrations: Second Order Systems with One Degree of Freedom, Free Response
Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single
More informationUse the following to answer question 1:
Use the following to answer question 1: On an amusement park ride, passengers are seated in a horizontal circle of radius 7.5 m. The seats begin from rest and are uniformly accelerated for 21 seconds to
More informationRotational Kinetic Energy
Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body
More informationLinear Second Order ODEs
Chapter 3 Linear Second Order ODEs In this chapter we study ODEs of the form (3.1) y + p(t)y + q(t)y = f(t), where p, q, and f are given functions. Since there are two derivatives, we might expect that
More informationMath 216 Final Exam 24 April, 2017
Math 216 Final Exam 24 April, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationPLANAR KINETIC EQUATIONS OF MOTION: TRANSLATION
PLANAR KINETIC EQUATIONS OF MOTION: TRANSLATION Today s Objectives: Students will be able to: 1. Apply the three equations of motion for a rigid body in planar motion. 2. Analyze problems involving translational
More informationWheel and Axle. Author: Joseph Harrison. Research Ans Aerospace Engineering 1 Expert, Monash University
Wheel and Axle Author: Joseph Harrison British MiddleEast Center for studies & Research info@bmcsr.com http:// bmcsr.com Research Ans Aerospace Engineering 1 Expert, Monash University Introduction A solid
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.0T Fall Term 2004 Problem Set 3: Newton's Laws of Motion, Motion: Force, Mass, and Acceleration, Vectors in Physics Solutions Problem
More informationFigure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m
LECTURE 7. MORE VIBRATIONS ` Figure XP3.1 (a) Mass in equilibrium, (b) Freebody diagram, (c) Kinematic constraint relation Example Problem 3.1 Figure XP3.1 illustrates a mass m that is in equilibrium and
More informationExam 2 March 12, 2014
Exam 2 Instructions: You have 60 minutes to complete this exam. This is a closedbook, closednotes exam. You are allowed to use a calculator during the exam. Usage of mobile phones and other electronic
More informationConcept Question: Normal Force
Concept Question: Normal Force Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is 1. larger than 2. identical
More informationModeling Mechanical Systems
Modeling Mechanical Systems Mechanical systems can be either translational or rotational. Although the fundamental relationships for both types are derived from Newton s law, they are different enough
More informationV sphere = 4 3 πr3. a c = v2. F c = m v2. F s = k s. F ds. = dw dt. P W t. K linear 1 2 mv2. U s = 1 2 kx2 E = K + U. p mv
v = v i + at x = x i + v i t + 1 2 at2 v 2 = v 2 i + 2a x F = ma F = dp dt P = mv R = v2 sin(2θ) g v dx dt a dv dt = d2 x dt 2 x = r cos(θ) V sphere = 4 3 πr3 a c = v2 r = rω2 F f = µf n F c = m v2 r =
More informationLab 11 Simple Harmonic Motion A study of the kind of motion that results from the force applied to an object by a spring
Lab 11 Simple Harmonic Motion A study of the kind of motion that results from the force applied to an object by a spring Print Your Name Print Your Partners' Names Instructions April 20, 2016 Before lab,
More informationProfessor Fearing EE C128 / ME C134 Problem Set 2 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C128 / ME C134 Problem Set 2 Solution Fall 21 Jansen Sheng and Wenjie Chen, UC Berkeley 1. (15 pts) Partial fraction expansion (review) Find the inverse Laplace transform of the following
More informationLaboratory handouts, ME 340
Laboratory handouts, ME 340 This document contains summary theory, solved exercises, prelab assignments, lab instructions, and report assignments for Lab 4. 20142016 Harry Dankowicz, unless otherwise
More informationMath 216 Second Midterm 20 March, 2017
Math 216 Second Midterm 20 March, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material
More informationInClass Problems 3032: Moment of Inertia, Torque, and Pendulum: Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 TEAL Fall Term 004 InClass Problems 303: Moment of Inertia, Torque, and Pendulum: Solutions Problem 30 Moment of Inertia of a
More informationSection 1 Simple Harmonic Motion. The student is expected to:
Section 1 Simple Harmonic Motion TEKS The student is expected to: 7A examine and describe oscillatory motion and wave propagation in various types of media Section 1 Simple Harmonic Motion Preview Objectives
More informationNoninertial Reference Frames
Chapter 12 Non Reference Frames 12.1 Accelerated Coordinate Systems A reference frame which is fixed with respect to a rotating rigid is not. The parade example of this is an observer fixed on the surface
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science
UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER 2016 EXAMINATIONS PHY 151H1F Duration  3 hours Attempt all 10 questions. All questions are worth 5 points. Write your name and student number
More information